C*-like modules and matrix p-operator norms

We present a generalization of H\"older duality to algebra-valued pairings via Lᵖ-modules. H\"older duality states that if p ∈ (1, ∞) and p^ are conjugate exponents, then the dual space of Lᵖ(μ) is isometrically isomorphic to L^p^(μ). In this work we study certain pairs (Y,X), as generalizations of the pair (L^p^(μ), Lᵖ(μ)), that have an Lᵖ-operator algebra valued pairing Y × X → A. When the A-valued version of H\"older duality still holds, we say that (Y,X) is C*-like. We show that finite and countable direct sums of the C*-like module $(A,A)$ are still C*-like when A is any block diagonal subalgebra of d × d matrices. We provide counterexamples when A ⊂ Mdᵖ(C) is not block diagonal.